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Analyzed the data: KV. Contributed reagents/materials/analysis tools: KV. Wrote the paper: KV.

The author has declared that no competing interests exist.

The idea that demographic change may spur or slow down technological change has become widely accepted among evolutionary archaeologists and anthropologists. Two models have been particularly influential in promoting this idea: a mathematical model by Joseph Henrich, developed to explain the Tasmanian loss of culture during the Holocene; and an agent-based adaptation thereof, devised by Powell et al. to explain the emergence of modern behaviour in the Late Pleistocene. However, the models in question make rather strong assumptions about the distribution of skills among social learners and about the selectivity of social learning strategies. Here I examine the behaviour of these models under more conservative and, on empirical and theoretical grounds, equally reasonable assumptions. I show that, some qualifications notwithstanding, Henrich’s model largely withstands my robustness tests. The model of Powell et al., in contrast, does not–a finding that warrants a fair amount of skepticism towards Powell et al.’s explanation of the Upper Paleolithic transition. More generally, my evaluation of the accounts of Henrich and of Powell et al. helpfully clarify which inferences their popular models do and not support.

Numerous recent publications

Bentley and O’Brien

Below, I do precisely that. I first show that Henrich’s results, although becoming more fragile, still obtain when relaxing (A.1), i.e. under assumptions of Normality. Second, I demonstrate that the population effect in Powell et al’s model vanishes when (A.2) takes on more conservative values, i.e. when mentor selection is assumed random or conformist. The lack of firm theoretical and empirical justification for (A.2), I argue, seriously compromises Powell et al’s explanation of the Upper Paleolithic transition.

Henrich’s mathematical model starts with a population of

In this equation, the variable

As it stands, Equation (1) is still not very helpful for studying the specific conditions under which

So if all imitators copy the most skilled individual,

The first part of Equation (2) takes the difference between the

Still assuming a Gumbel distribution of skills,

The second part of Equation (2), namely

Since

Substituting Equations (4), (5) and (6) in Equation (2) yields

Setting

For the Gumbel,

I must add here that Henrich also considers what happens in case skills follow a standard Logistic distribution. Following the same derivation as above, transmission inaccuracy for the Logistic,

The two variants of the model indeed support the idea idea that only sizable populations of social learners are able to sustain processes of technological accumulation. To see whether this result is robust when we relax (A.1), the model now needs to be adjusted to assumptions of Normality.

Henrich provides no justification for using Logistic distributions. For the Gumbel, in contrast, his justification is that “a wide range of distributions, which includes the Gumbel distribution and the Normal distribution, yield the Gumbel distribution (approximately) when the extreme values are repeatedly sampled”

To be clear, whether skills of the sort targeted by Henrich’s model (e.g., arrow-making, fishtrap-building) follow a Gumbel, Logistic or Normal distribution is an open empirical question. Gumbel distributions are typically used to account for extreme events (e.g., extraordinary hurricanes, floods, winds, earthquakes). Perhaps, intuitively, one may think that this matches cases where technological skills are taught by an extremely gifted tutor, say, a grandmaster training several novice archers. In such cases, at least when the population is considered as a whole (including tutors

How do we enter assumptions of Normality in Henrich’s model? We start with Equation (2) again, but redefine the variables at the right-hand side of the equation to reflect the idea that skill levels are distributed Normally. The expected value of the highest values drawn from a sample of size

The other terms in Equation (2) are redefined as follows:

Based on Henrich’s work, Powell et al

Inspired by Henrich, Powell et al assume: (A.1) Gumbel rather than Normal distributions of skills; and (A.2) a strong tendency to select the most skilled individuals in the population as mentors. In contrast to Henrich, however, Powell et al wish to explain an instance of cultural

The model of Powell and colleagues contains a set of agents that undergo both vertical and oblique transmission. In particular, in each generation the model goes through the following steps:

At this point, two things need to be stressed. First, Powell et al’s original model contains also a fourth step, in which agents are given the opportunity to migrate from one subpopulation to the other. But since the authors find that migratory activity among a set of subpopulations has the same effect on skill accumulation as increasing the size of a single isolated population, this step is ignored in the remainder. That is, my adjusted model will only concern isolated populations with varying sizes

Second, the strong selectivity assumption (A.2) is made in step 2, in the passage in italics: adults choose mentors proportionally to their skill. This is a somewhat weaker form of selectivity than that assumed by Henrich in Equation (7), but still much stronger than the two I will consider, namely random copying and conformity.

In the random copying condition, offspring select an oblique model at random and adopt the latter’s

Such randomness may reflect two things. First, it may capture the fact that, especially in larger populations and/or populations with low cultural interconnectedness, not all cultural parents are available to all to learn from, so that happenstance decides which parents are assigned to whom. Second, the random copying condition may imply absence of learning biases or, if present, any systematic expression of them.

As Bentley and colleagues remark

Moreover, Powell et al’s selectivity is at odds with the conformity biases widely discussed in studies of cultural evolution. Conformity here refers to the propensity of individuals to

For given values of

As can be seen in

The above may seem to prove wrong Bentley and O’Brien

More specifically, Normal populations are less susceptible to loss due to population reductions than Gumbel/Logistic populations. To appreciate this, one just needs to compare the derivatives of the functions governing the Gumbel/Logistic and Normal curves presented in

Comparison is done by plotting the derivative functions of, on one hand,

To stress again, these differences are subtle, and arguably insufficient to undermine Henrich’s explanation of the loss of Tasmanian culture (although the matter is impossible to settle empirically). Yet, what stands out is that demographic explanations under assumptions of Normality are somewhat more demanding: to account for an observed loss of culture, bigger population reductions must be assumed.

Henrich’s model was devised explicitly to address instances of cultural

However, in explanations of cultural

Hence, without modifying its assumptions (i.e. assuming much-less-than-perfect mentor selection), Henrich’s model is likely to overestimate the cultural gain resulting from increases in population size (unless, of course, the assumption of perfect mentor selection really would conform with empirical observations). Importantly, Henrich himself doesn’t use his model for that purpose (in contrast to Powell et al, see below). Moreover, he presents an adjustment of Equation (7) (see his Equation 4) which can be used to examine the effects of random copying (although it is presented as an adjustment to accommodate vertical transmission). In case all individuals copy at random, the model now predicts no cumulation, which is more or less in line with the results presented in the next section. It is unclear how Henrich’s model would behave under conditions of strong conformity, but these will be taken up in the next section.

The considerations above apply irrespective of distribution choice: neither the Gumbel/Logistic nor the Normal implementation of Henrich’s model are duly conservative when it concerns demographic explanations of instances of technological gain. But, the Normalized model is even worse in this respect. For, given the more modest population effect under assumptions of Normality (see result b), Normal populations will even more readily exhibit cultural stasis, despite increases of

For both (A) and (B),

There are indeed theoretical reasons for assuming pay-off biases

In light of the above, I think the most honest thing to say is that the issue is undecided: there is insufficient warrant for choosing either bias (pay-off or conformist) as being the most dominant in human social learning, let alone as being the most dominant in Upper Paleolithic cultural learning. Hence, Powell et al have little warrant for focusing only on the most optimistic scenario (strong pay-off biased transmission), simply ignoring the more conservative one (strong conformist transmission).

Second, what about the random copying condition? On the first interpretation of the condition, randomness has nothing to do with the psychological make-up of the social learners in question, but with the fact that external factors often do the selection for us, for instance, when populations are big and/or exhibit low cultural interconnectedness. Chance effects due to population size are presumably limited for the sizes assumed by Powell et al (

According to the second interpretation, the random copying condition implies randomness in the strategies of social learners when selecting a mentor. The learner is assumed either to really select mentors at random (i.e. she is insensitive to clues about such things as the mentor’s success or the popularity of the mentor’s behaviour), or to switch strategies seemingly at random (e.g, conformist under time pressure, pay-off based otherwise; pay-off based if cheap, conformist otherwise; conformist in the production of threaded shell beads and tattoo kits, pay-off based in matters of blades and burins). While there is accumulated evidence that contemporary humans do not just select mentors at random (for a recent report, see

In sum, given that Powell et al’s results do not survive assumptions of weak selectivity, the authors need to provide a justification for the strong pay-off biases they assume. In light of the evidence just discussed, this will be a difficult, if not impossible, task.

As can be seen in

In this paper, I have examined the robustness and explanatory power of two very influential models linking demography and cumulative culture, a mathematical model by Henrich, and an agent-based implementation thereof by Powell and colleagues. In particular, I have tested Bentley and O’Brien’s suggestion that the results obtained by Henrich and by Powell et al may be an artifact of two fairly strong assumptions: (A.1) that cultural skill levels follow a Gumbel (or a Logistic, in case of Henrich) rather than a Normal distribution; and (A.2) that social learning biases are strongly pay-off based. Relaxing (A.1) and (A.2), I found, has limited impact on Henrich’s account, but seriously comprises that of Powell et al.

More specifically, Henrich’s model still exhibits a population effect if (A.1) is relaxed, although adaptive culture in Normal populations appears generally less sensitive to population reduction than adaptive culture in Gumbel/Logistic populations. Relaxing (A.2), as anticipated by Henrich himself, potentially removes the effect completely. But since Henrich’s aim is to explain an instance of cultural loss, (A.2), even if unrealistic, provides the severest test possible. Loss demonstrated under (A.2) would be even more likely in case learning biases were less pay-off based.

Powell et al’s model, in contrast, targets an instance of cultural gain. Here severe tests are those in which (A.2)

The present study is limited in scope in that it has focused on just two, even if very influential, models. Future research should determine whether alternative models linking population size and cultural adaptiveness

Details of the implementation of the simulations.

(PDF)

Krist Vaesen would like to thank Wybo Houkes and four anonymous reviewers for their useful comments on previous versions of this paper.